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In 1973, Graham and Kleitman proved that $f(K_n) \\ge \\sqrt{n - 3/4} - 1/2$ and in 1984, Calderbank, Chung, and Sturtevant proved that $f(K_n) \\le (\\frac{1}{2} + o(1))n$. We show that $f(K_n) \\ge (\\frac{1}{20} - o(1))(n/\\lg n)^{2/3}$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1509.02143","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-07T19:08:14Z","cross_cats_sorted":[],"title_canon_sha256":"82abca5a4deb8d5bdbfa9eb150bbe7bdf46489dce57be7dc6ed9d1b2ab685127","abstract_canon_sha256":"b2f9cc8e404d6cfed7a48f379cdf838691003a6e029ca78130498927939b27f4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:49.642053Z","signature_b64":"QIxny47WiFRhyYLK4tIvUTG3djpqXVFZdUx7BKNv7cDbbgQRTgjg2cYHwwVXzEfB16bLm5AvApKgqVNP5wh9Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"386721a6351d4bfe5c1a01584d5441cc3179b72811a9e013651db0cf296aa889","last_reissued_at":"2026-05-18T01:33:49.641461Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:49.641461Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Monotone Paths in Dense Edge-Ordered Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kevin G. Milans","submitted_at":"2015-09-07T19:08:14Z","abstract_excerpt":"The altitude of a graph $G$, denoted $f(G)$, is the largest integer $k$ such that under each ordering of $E(G)$, there exists a path of length $k$ which traverses edges in increasing order. In 1971, Chv\\'atal and Koml\\'os asked for $f(K_n)$, where $K_n$ is the complete graph on $n$ vertices. In 1973, Graham and Kleitman proved that $f(K_n) \\ge \\sqrt{n - 3/4} - 1/2$ and in 1984, Calderbank, Chung, and Sturtevant proved that $f(K_n) \\le (\\frac{1}{2} + o(1))n$. We show that $f(K_n) \\ge (\\frac{1}{20} - o(1))(n/\\lg n)^{2/3}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.02143","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1509.02143","created_at":"2026-05-18T01:33:49.641542+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.02143v1","created_at":"2026-05-18T01:33:49.641542+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.02143","created_at":"2026-05-18T01:33:49.641542+00:00"},{"alias_kind":"pith_short_12","alias_value":"HBTSDJRVDVF7","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_16","alias_value":"HBTSDJRVDVF74XA2","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_8","alias_value":"HBTSDJRV","created_at":"2026-05-18T12:29:22.688609+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HBTSDJRVDVF74XA2AFME2VCBZQ","json":"https://pith.science/pith/HBTSDJRVDVF74XA2AFME2VCBZQ.json","graph_json":"https://pith.science/api/pith-number/HBTSDJRVDVF74XA2AFME2VCBZQ/graph.json","events_json":"https://pith.science/api/pith-number/HBTSDJRVDVF74XA2AFME2VCBZQ/events.json","paper":"https://pith.science/paper/HBTSDJRV"},"agent_actions":{"view_html":"https://pith.science/pith/HBTSDJRVDVF74XA2AFME2VCBZQ","download_json":"https://pith.science/pith/HBTSDJRVDVF74XA2AFME2VCBZQ.json","view_paper":"https://pith.science/paper/HBTSDJRV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.02143&json=true","fetch_graph":"https://pith.science/api/pith-number/HBTSDJRVDVF74XA2AFME2VCBZQ/graph.json","fetch_events":"https://pith.science/api/pith-number/HBTSDJRVDVF74XA2AFME2VCBZQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HBTSDJRVDVF74XA2AFME2VCBZQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HBTSDJRVDVF74XA2AFME2VCBZQ/action/storage_attestation","attest_author":"https://pith.science/pith/HBTSDJRVDVF74XA2AFME2VCBZQ/action/author_attestation","sign_citation":"https://pith.science/pith/HBTSDJRVDVF74XA2AFME2VCBZQ/action/citation_signature","submit_replication":"https://pith.science/pith/HBTSDJRVDVF74XA2AFME2VCBZQ/action/replication_record"}},"created_at":"2026-05-18T01:33:49.641542+00:00","updated_at":"2026-05-18T01:33:49.641542+00:00"}